Let R be a commutatuve ring with an identity. a.) Fix an element a in R and cons

ID: 2963237 • Letter: L

Question

Let R be a commutatuve ring with an identity.

a.) Fix an element a in R and consider the function "multiplication by a" defined formally as

(m: R ightarrow R ext {, where }m(x)=ax.)

Prove that the function m is a ring homomorphism if and only if a * a = a............(hint 1*1=1)

b.) If R is an integral domain, how many such "multiplication homomorphisms" are there? what are they and why are there no more?

c.) What is an example of a commutative ring with identity with at least three such multiplication homomorphisms?

I have written the answer on a paper and uploaded it as 4 pictures at the following links:

PArt-1: http://i.imgur.com/btCIP6R.jpg

Part-2: http://i.imgur.com/3MVn5tF.jpg

Part-3: http://i.imgur.com/FWCjSa3.jpg

Part-4: http://i.imgur.com/dykr7us.jpg

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